We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Anderson, A., and N. Belnap, Entailment, vol. 1, Princeton University Press, 1975.
Arieli O., Avron A.: ‘The value of the four values’. Artificial Intelligence 102(1), 97–141 (1998)
Arieli O., Avron A., Zamansky A.: ‘Maximal and premaximal paraconsistency in the framework of three-valued semantics’. Studia Logica 97(1), 31–60 (2011)
Arieli, O., A. Avron, and A. Zamansky, ‘What is an ideal logic for reasoning with inconsistency?’, in Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11), AAAI Press, 2011, pp. 706–711.
Avron A.: ‘Natural 3-valued logics: Characterization and proof theory’. Journal of Symbolic Logic 56(1), 276–294 (1991)
Avron, A., O. Arieli, and A. Zamansky, ‘On strong maximality of paraconsistent finite-valued logics’, in Proceedings of the 25th Ann. Symp. on Logic in Computer Science (LICS’10), IEEE Press, 2010, pp. 304–313.
Avron A., Lev I.: ‘Non-deterministic multi-valued structures’. Journal of Logic and Computation 15, 241–261 (2005)
Avron, A., and A. Zamansky, ‘Non-deterministic semantics for logical systems – A survey’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, Kluwer, 2011. To appear.
Batens D.: ‘Paraconsistent extensional propositional logics’. Logique et Analyse 90-91, 195–234 (1980)
Batens, D., C. Mortensen, G. Priest, and J. Van Bendegem, Frontiers of Paraconsistent Logic, Proceedings of the First World Congress on Paraconsistency, vol. 8 of Studies in Logic and Computation, Research Studies Press, 2000.
Belnap, N., ‘How a computer should think’, in G. Ryle, (ed.), Contemporary Aspects of Philosophy, Oriel Press, 1977, pp. 30–56.
Belnap, N., ‘A useful four-valued logic’, in J. M. Dunn, and G. Epstein, (eds.), Modern Uses of Multiple-Valued Logics, Reidel Publishing Company, 1977, pp. 7–37.
Béziau, J. Y., W. Carnielli, and D. Gabbay, Handbook of Paraconsistency, 2007.
Bremer, M., An Introduction to Paraconsistent Logics, Peter Lang, 2005.
Carnielli, W., M. Coniglio, and I. D’Ottaviano, Paraconsistency: The Logical Way to the Inconsistent – Proceedings of the Second World Congress on Paraconsistency, Marcel Dekker, 2001.
Carnielli, W., M. Coniglio, and J. Marcos, ‘Logics of formal inconsistency’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 14, Springer, 2007, pp. 1–93. Second edition.
Carnielli W., Marcos J., de Amo S.: ‘Formal inconsistency and evolutionary databases’. Logic and Logical Philosophy 8, 115–152 (2000)
da Costa N.: ‘On the theory of inconsistent formal systems’. Notre Dame Journal of Formal Logic 15, 497–510 (1974)
D’Ottaviano I.: ‘The completeness and compactness of a three-valued first-order logic’. Revista Colombiana de Matematicas XIX(1-2), 31–42 (1985)
Dunn J.M.: ‘Intuitive semantics for first-degree entailments and coupled trees’. Philosophical Studies 29, 149–168 (1976)
Fitting M.: ‘Bilattices and the semantics of logic programming’. Journal of Logic Programming 11(2), 91–116 (1991)
Fitting M.: ‘Kleene’s three valued logics and their children’. Fundamenta Informaticae 20(1-3), 113–131 (1994)
Ginsberg M.: ‘Multi-valued logics: A uniform approach to reasoning in AI’. Computer Intelligence 4, 256–316 (1988)
Gottwald, S., ‘A treatise on many-valued logics’, in Studies in Logic and Computation, vol. 9, Research Studies Press, Baldock, 2001.
Jaśkowski, S., ‘On the discussive conjunction in the propositional calculus for inconsistent deductive systems’, Logic, Language and Philosophy 7:57–59, 1999. Translation of the original paper from 1949.
Karpenko, A., ‘A maximal paraconsistent logic: The combination of two threevalued isomorphs of classical propositional logic’, , pp. 181–187.
Kleene, S. C., Introduction to Metamathematics, Van Nostrand, 1950.
Malinowski, G., Many-Valued Logics, Clarendon Press, 1993.
Marcos, J., ‘8K solutions and semi-solutions to a problem of da Costa’, Submitted.
Marcos, J., ‘On a problem of da Costa’, in G Sica, (ed.), Essays on the Foundations of Mathematics and Logic, vol. 2, Polimetrica, 2005, pp. 39–55.
Priest G.: ‘Logic of paradox’. Journal of Philosophical Logic 8, 219–241 (1979)
Priest G.: ‘Reasoning about truth’. Artificial Intelligence 39, 231–244 (1989)
Sette A.M.: ‘On propositional calculus P 1’. Mathematica Japonica 16, 173–180 (1973)
Shoesmith D.J., Smiley T.J.: ‘Deducibility and many-valuedness’. Journal of Symbolic Logic 36, 610–622 (1971)
Shoesmith, D. J., and T. J. Smiley, Multiple Conclusion Logic, Cambridge University Press, 1978.
Urquhart, A., ‘Many-valued logic’, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. II, Kluwer, 2001, pp. 249–295. Second edition.
Dedicated to Professor Ryszard Wójcicki on the occasion of his 80th birthday
Special issue in honor of Ryszard Wójcicki on the occasion of his 80th birthday
Edited by J. Czelakowski, W. Dziobiak, and J. Malinowski
About this article
Cite this article
Arieli, O., Avron, A. & Zamansky, A. Ideal Paraconsistent Logics. Stud Logica 99, 31 (2011). https://doi.org/10.1007/s11225-011-9346-y
- Paraconsistent logics
- ideal paraconsistency
- many-valued logics